Planar-Toroidal Decomposition of $K_{12}$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918381073989632 |
|---|---|
| author | Bickle, Allan Campbell, Russell |
| author_facet | Bickle, Allan Campbell, Russell |
| contents | In 1978, Anderson and White asked whether there is a decomposition of $K_{12}$ into two graphs, one planar and one toroidal. Using theoretical arguments and a computer search of all maximal planar graphs of order 12, we show that no such decomposition exists. We further show that if $G$ is planar of order 12 and $H\subseteq\overline{G}$ is toroidal, then $H$ has at least two fewer edges than $\overline{G}$. A computer search found all 123 unique pairs $\left(G,H\right)$ that make this an equality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_17084 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Planar-Toroidal Decomposition of $K_{12}$ Bickle, Allan Campbell, Russell Combinatorics 05C10 In 1978, Anderson and White asked whether there is a decomposition of $K_{12}$ into two graphs, one planar and one toroidal. Using theoretical arguments and a computer search of all maximal planar graphs of order 12, we show that no such decomposition exists. We further show that if $G$ is planar of order 12 and $H\subseteq\overline{G}$ is toroidal, then $H$ has at least two fewer edges than $\overline{G}$. A computer search found all 123 unique pairs $\left(G,H\right)$ that make this an equality. |
| title | Planar-Toroidal Decomposition of $K_{12}$ |
| topic | Combinatorics 05C10 |
| url | https://arxiv.org/abs/2507.17084 |